Product of nullhomologous curve with $S^{1}$ factor is still nullhomologous

Question

Let $N$ be a nullhomologous curve in a $3$-manifold $X$. Let $S^1\times X$ be the product manifold.

Why is then $S^1\times N$ nullhomologous in $S^1\times X$?

PS: Is nullhomologous just a statement about the class in the first homology group or in all homology groups?

Answer 1

Let $i : S^1 \to X$ be a closed curve $N$. We say $N$ is nullhomologous in $X$ if $i_*[S^1] \in H_1(X)$ is zero, where $[S^1]$ is the the fundamental class of $S^1$. Likewise, we say $S^1\times N$ is nullhomologous in $S^1\times X$ if $(\operatorname{id}_{S^1}, i)_*[S^1\times S^1] \in H_2(S^1\times X)$ is zero.

Under the Künneth isomorphism $H_2(S^1\times X) \cong H_1(S^1)\otimes H_1(X) \oplus H_2(X)$, $(\operatorname{id}_{S^1}, i)_*[S^1\times S^1] \mapsto (\operatorname{id}_{S^1})_*[S^1]\otimes i_*[S^1]$.

So if $N$ is nullhomolgous in $X$, $i_*[S^1] = 0$, so $(\operatorname{id}_{S^1}, i)_*[S^1\times S^1] = 0$ and hence $S^1\times N$ is nullhomologous in $S^1\times X$.

In fact, as $(\operatorname{id}_{S^1})_*[S^1] = [S^1] \neq 0$, $S^1\times N$ is nullhomologous in $S^1\times X$ if and only if $N$ is nullhomologous in $X$.